19th century: The American expansion was guided by the concept of <em>Manifest Destiny</em>, being that the people of the time believed it was their fate to expand and colonize the rest of the territory (that became what is now the U.S.), whilst pushing forward their virtues and institutions, with the urge to do so being irresistible to them.
20th century: The expansion of this period (that actually started in the final years of the century before) was called <em>Imperialism</em>, where the idea of gaining overseas territories, expanding American influence on international market by expanding their industry and trade.
Similarities and differences: In both periods there was an interest in expanding American territories, although the ideologies behind those movements where different: in the former the belief of forming a great country through force of will was their core motive; conflicts with other nations and cultures were consequences rather than the motif. In the later the economic and power interest where the reasons for doing so; the expansion had many morally questionable sub-tones, such as racism and an exaggerated me-before-you approach to all, with conflict and war being promoted by one president of the time (Theodore Roosevelt).
